Van Vleck’s formula

Consider the orbital angular momentum of a free-ion term. Here L = 3 and the orbital degeneracy is 7. Application of Van Vleck s formula (5.8) predicts an effective magnetic moment. 

In octahedral symmetry, the F term splits into Aig + T2g + Tig crystal-field terms. Suppose we take the case for an octahedral nickel(ii) complex. The ground term is 2g. The total degeneracy of this term is 3 from the spin-multiplicity. Since an A term is orbitally (spatially) non-degenerate, we can assign a fictitious Leff value for this of 0 because 2Leff+l = 1. We might employ Van Vleck s formula now in the form... 

An instructive illustration of the effect of molecular motion in solids is the proton resonance from solid cyclohexane, studied by Andrew and Eades 101). Figure 10 illustrates their results on the variation of the second moment of the resonance with temperature. The second moment below 150°K is consistent with a Dsi molecular symmetry, tetrahedral bond angles, a C—C bond distance of 1.54 A and C—H bond distance of 1.10 A. This is ascertained by application of Van Vleck s formula, Equation (17), to calculate the inter- and intramolecular contribution to the second moment. Calculation of the intermolecular contribution was made on the basis of the x-ray determined structure of the solid. 

On taking into consideration the electric dipoles as isotropically polarizable, we get in place of Van Vleck s formula (260) the relation ... 

The susceptibility does not obey the Curie-Weiss law above ca. 500 K because the value of the multiplet splitting is comparable with kT. Values of Xmoi calculated from Van Vleck s formula for the free Ce ion for various values of the screening constant (a = 34, 35, 36) fit the experimental Xmoi values well [8]. 

Fig. 50. Inverse magnetic susceptibility 1/Xmoi of NdSe and NdTe versus temperature up to 1300 K. The dashed curves are calculated from Van Vleck s formula for different screening constants a. The inset shows the range from 4 to 150 K in an extended temperature scale.

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

Van Vleck s second moment formula has been experimentally proven to be correct by Fake and Purcell (74)- It has become a valuable tool in structure determination of the solid state. 

Toupin and Lax consider the problem of permanent and induced dipoles on cubic lattice sites (or continuum) with the latter represented by harmonic oscillators as in Van Vleck s early work described in 1 3 The device of introducing fluctuations from equilibrium displacements works for harmonic oscillators because the integrations over the formula to evaluate averages are for all values from oO to and unchanged by the shifts in origin A similar device is not possible for proper averages over possible permanent dipole moments as the ranges are restricted by the N constraints becomes so if these are replaced by the... 

The application of angular-momentum theory to atomic spectroscopy is not limited to bringing eqs. (3)-(7) into play. In their book, Condon and Shortley (1935, ch. 3) developed the theory with particular attention to operators T that are vectors. They did this by specifying the commutation relations of Twith respect to J rather than by stating the transformation properties of the components of T under rotations. They considered angular momenta J built from two parts S and L) and obtained formulas for the matrix elements of operators that behave as a vector with respect to one part (say L) and a scalar with respect to the other (S, in this case). These formulas involve proportionality constants that would be called reduced matrix elements today. Condon and Shortley systematized the methods that had come into current use but which were often only hinted at, if that, by many theorists. For example. Van Vleck (1932) gave the formula... 

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